Other essential refs which seem to have been overlooked on this thread — the works of Peter & Iona Opie, 1950s-60s; esp The Lore & Language of Schoolchildren [excellent on all the children's variants on Disney's Davy Crockett song of 1956, also inexplicably overlooked], Children's Games In Street & Playground, Oxford Dict of Nursery Rhymes - all of which replete with examples of subject of this thread...

One which we haven't had so far which comes to mind is: 1 2 3 a-lairie, I saw Wallace Beery, Sitting on his bumbaleery, Kissing Shirley Temple [skipping or ball bouncing]. Also the Redwing variants about the sun shining bright on Charlie Chaplin...

Re Grand Old Duke of York: people in Barnet, North London, including my sister at one time, think it much older, relating to Barnet Hill & the Duke of York who fought the Battle of Barnet 1471.

The King George whose queen was called Mary, refd in above, was George V, 1910-36, grandfather of present Queen Elizabeth II.